Draco probably became a death eater because of the pressure from his mother and father. Being a deatheater is what was expected from him. Also, there was also the fact that if he DIDNT become one, he would be killed or something like that. Draco was probably one of the only people who could kill Dumbledore easily enough. He had easy access - they were often in the same room.

Another thing was probably because Draco wanted to be accepted among the people who he happens to look up to - Death Eaters. He no longer wanted to be a child, and the only way he could "belong" and have respect was if he killed Dumbledore.

It might have been important to belong, and the only place he thought he could belong was with Voldemort. Maybe he didn't really want to be one, but he was afraid that he couldn't be trusted or accepted in either Dumbledore's group, or even just as a normal wizard.

Hmm... was this confusing? Oh well, sorry about that.